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\begin{document}



\title{THE THEORY OF PLANE WAVE DIFFRACTION BY GRATINGS}
\author{A.G.Kyurkchan}
{Moscow Technical University of
Communication and Information Science \\
Aviamotornaya 8a, Moscow, 105855, Russia\\
Home address: Staromonetnii by-street 24, apt 15,
Moscow, 109180, Russia}
\coauthor{A.I.Kleev}
{P.Kapitza
Institute for Physical Problems \\
Kosygina 2, Moscow, 117334, Russia.\\
Fax: 7(095) 9382030 \\
E-mail: kleev@kapitza.ras.ru}


This complex, time consuming problem is of great interest to researches in
optics and antenna technology. Currently available methods of approaching the
problem are either quite specific or cumbersome and inefficient. Here we are
discussing an approach based on the pattern equation method \cite{1,2}. An
algorithm is developed, which can be used for calculation of diffraction
grating characteristics with rather loose restrictions both on the geometry
of individual elements and their closeness. With this approach the
diffraction gratings can be linear (one-dimensional or planar,
two-dimensional).

Using the pattern equation method an original boundary problem can be reduced
to the set of algebraic equations relative to the Fourier coefficients for
central element scattering patterns as follows
$$
a_{pm}=a_{pm}^0+\sum\limits_{s=1}^N\sum\limits_{n=-\infty}^\infty
G_{mn}^{ps}a_{sn};\quad p=1,2,\dots,N;\quad
m=0,\pm 1,\dots
$$
where
$a_{pm}$ is the
$m$-th Fourier coefficient for the central element of
$p$-th grating;
$a_{pm}^0$
are known quantities depending on the primary plane wave angle of incidence
$\varphi_0$;
$G_{mn}^{ps}$ are matrix elements.
Analyzing asymptotic of the matrix elements
$G_{mn}^{ps}$
and free terms
$a_{pm}^0$ with the proviso that
$|m|\gg 1$, $|n|\gg 1$
one can set forth the constraints on the grating central element
cross-section geometries as well as the closeness of gratings and their
individual elements when the set of equations can be solved using the
reduction method. This constraints are not stringent. So, for example, all
the convex contours and so-called weakly non-convex contours \cite{1} are
rated in the class of acceptable cylindrical element cross-section contours.
Grating elements can be arranged arbitrarily close to each other to the
point of contacting.
The proposed algorithm can be reduced to fairly simple and
demonstrable equations which can be used to obtain explicit approximate
formulas for various kinds of asymptotic. For example, given the Direchlet's
boundary condition and
$a/\lambda\ll 1$, $b/\lambda\ll 1$
(where
$b$ is the period of grating and
$\lambda$ is the wavelength)
the following formula can be derived
$$
R_0\approx -\left[1-i\frac{2b}{\lambda}\sin\varphi_0
\ln\frac{2\pi a}{b}\right]^{-1}
$$
The formula for the reflection coefficient of double grating composed of
round cylinders, derived to this approximation, agrees (with a precision of
symbols) with equivalent formula obtained through the use of the induced EMF
method. The algorithm is noted for its very fast convergence and can be
generalized to problems of gratings in layered medium as well as to
three-dimensional problems

\begin{thebibliography}{9}
\bibitem{1} Kyurkchan A.G. //Proceedings of the 1995 URSI Intern. Symp.
on Electromagnetic Theory. St. Petersburg, 1995, p. 659.
\bibitem{2} Kyurkchan A.G. //Journal of Communications Technology
and Electronics, {\bf 41}(1), pp. 40--45 (1996).
\end{thebibliography}

\end{document}
